phys. stat. sol. (a) 203, No. 6, 1154–1159 (2006) / DOI 10.1002/pssa.200566113

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Photon-assisted tunneling

in ac driven double quantum dot spin pumps

R. Sánchez*, 1, G. Platero1, R. Aguado1, and E. Cota2

1 Instituto de Ciencia de Materiales de Madrid-CSIS, Cantoblanco, 28049, Spain 2 Centro de Ciencias de la Materia Condensada-UNAM, Ensenada B.C., México

Received 22 September 2005, revised 8 January 2006, accepted 8 January 2006

Published online 3 May 2006

PACS 73.23.Hk, 73.63.Kv, 85.75.–d

In this work, we study the effect of an applied ac gate voltage on the spin filtering and pumping properties

of a lateral double quantum dot, in the Coulomb blockade and weak coupling regimes, considering not

only the effect of the ac potential on the inter-dot tunneling but also on the tunneling through the leads.

This last effect accounts for additional photon absorption and emission processes and therefore affects the

spin polarization of the pumped current. In particular, we find that the spin down filtering property can be

affected depending on the intensity of the ac field.

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The spin degree of freedom of the electron and its transport properties through mesoscopic systems are

responsible for a new field of research, called spintronics, where the spin determines the physical proper-

ties as does the charge in conventional electronics. Specifically, the long spin decoherence and relaxation

times in semiconductor materials and the possibility of manipulating single electrons in quantum dots

(QD) have generated great interest because of the fundamental physics involved and also for the eventual

development of technological devices.

The control and manipulation of electron spins, is one of the main tasks in the development of solid

state based quantum computation [1, 2]. In that way, recently, experiments based on spin to charge con-

version of a single electron confined in a QD have been proposed to read-out spin single electrons [3].

Moreover, QD systems have been operated as bipolar spin filters [4].

Recently, it has be shown [5, 6] that it is possible to create spin-polarized current in a double quantum

dot (DQD) system driven by an ac potential applied between the dots in the presence of a uniform mag-

netic field [5–7] in the pumping configuration, i.e., with no bias voltage applied to the leads. In these

works the effect of the ac potential is considered only to affect the inter-dot tunneling but not the tunnel-

ing between leads and dots [6, 11]. In order to evaluate how this affects the spin transport properties of

the device, in the present work, we extend previous models to include the effect of possible transitions

due to photon emission or absorption at the leads.

2 Theoretical model

Our system consists on an asymmetric double quantum dot (DQD) connected in series to two leads and one

to each other by tunnel barriers. We describe it with the Hamiltonian ˆH = L

ˆH + R

ˆH + L R

ˆH¤

+ leads

ˆH + T

ˆH .

* Corresponding author: e-mail: rafael.sanchez@icmm.csic.es, Phone: +34 913 721 420, Fax: +34 913 720 623

phys. stat. sol. (a) 203, No. 6 (2006) 1155

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Paper

We model each quantum dot as a one-level Anderson impurity: †L R

ˆ

ˆ ˆˆ ˆj j jj j j jH U n nc cσ σσ

σ

ε= , Ø≠= + , where

the operator †ˆ jc σ

creates an electron with spin σ in dot j with an energy jσε and jU is the charging en-

ergy of each dot. We also introduce a magnetic field in order to break the spin degeneracy of the different

levels by a Zeeman splitting, j∆ . We consider the spin-up state as the ground state, so that jj jε ε ∆

Ø Øfi + .

The Hamiltonian term corresponding to the leads is written as: †

leads

L R

ˆ ˆˆ

lk lk lk

l k

H d dσ σ

ε σ

ε

,

= Â , while the

inter-dot coupling is: †

L R LR L Rˆ ˆ ˆ h.c.H t c c

σ σ

σ

¤= - +Â We treat the coupling of the DQD to the leads

†

T

L R

ˆˆ ˆ( h.c.)l lk l

l k

H d cσ σ σ

ε σ

γ

,

= +Â as a perturbation. We assume the wide-band limit, i.e., lσ

γ is energy inde-

pendent. The pumping of electric current is based on the introduction of an external ac field acting on the dots:

L/R L/R L/R L/R, acˆ ˆ ˆ ˆ( ) ( )H H t H H tÆ = + , (1)

from which:

acL(R ) L(R ) L(R )( ) ( ) cos

2i

Vt t

σ σ σε ε ε ωÆ = + - , (2)

(in units where 1 1e= , = ) where ac

V and ω are the amplitude and frequency of the applied field. Whith

this, the system has now an explicit time dependence. The time dependent field may affect, as we will

show below, not only the inter-dot tunneling but also the tunneling through the leads [13].

We consider up to four extra electrons in the system, then the basis consists of 16 states, as follows:

|1 |0 0Ò = , Ò , |2 | 0Ò = ≠, Ò, |3 | 0Ò = Ø, Ò, |4 |0Ò = , ≠Ò , |5 |0Ò = , ØÒ, |6 |Ò = ≠, ≠Ò , |7 |Ò = Ø, ØÒ , |8 |Ò = ≠, ØÒ, |9 |Ò = Ø, ≠Ò , |10 | 0Ò = ≠Ø, Ò , |11 |0Ò = , ≠ØÒ , |12 |Ò = ≠Ø, ≠Ò , |13 |Ò = ≠Ø, ØÒ , |14 |Ò = ≠, ≠ØÒ , |15 |Ò = Ø, ≠ØÒ ,

|16 |Ò = ≠Ø, ≠ØÒ .

2.1 Master equation

In order to study the electron dynamics of the DQD connected to reservoirs we have considered the re-

duced density operator, ˆ ˆ

Rtrρ χ= , where one traces over all the reservoir states in the complete density

matrix of the system, χ . The evolution of the time independent system will be given by the Liouville

equation: ˆd ( ) ˆ ˆ[ ( )]d

ti H t

t

ρρ= - , . Assuming the Markov approximation [8], we obtain the master equation,

written as:

ˆ ˆ( ) ( ) [ ( )]m m m m m m L R m m

t i t i H tρ ω ρ ρ¢ ¢ ¢ ¤ ¢= - - ,

( ) ( )m n nn nm mm m m

n m n m

t tΓ ρ Γ ρ δ¢ ¢

π ¢ π

Ê ˆ+ -Á ˜Ë ¯Â  ( ) (1 )m m m m m m

tΩ ρ δ¢ ¢ ¢

- - , (3)

where m m

ω¢

is the energy difference between the states |m Ò¢ and |mÒ of the isolated DQD and m m

Ω¢

de-

scribes the decoherence of the DQD states due to the interaction with the reservoir. mn

Γ are the transition

rates for electrons tunneling through the leads for the time independent system, which have the usual

Fermi Golden Rule expression:

2

1 12π | | ( ( ) (1 ( )) )

m n m nmn mn mn l N N nm l N N

l

f fΓ γ ω µ δ ω µ δ, + , -

= - + - - , (4)

where L Rl = , denotes the leads and l

µ are the corresponding chemical potentials. k

N is the number of

electrons in the system in state |kÒ , mn

γ is the matrix element of T

ˆH connecting the states |mÒ and |nÒ and ( )( ) 1 (1 e )f ε µ β

ε µ-

- = / + is the Fermi distribution function, where B

1 k Tβ = / . These transition rates

1156 R. Sánchez et al.: Photon-assisted tunneling in ac driven double quantum dot spin pumps

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-a.com

(Eq. (4)) are related to the decoherence through the relation: 1

2m m km km

k m k m

Ω Γ Γ

Ê ˆÁ ˜Á ˜¢ ¢Á ˜Ë ¯π ¢ π

¬ = +Â Â . If we now

introduce the oscillation of the energy levels (2): ( )m m m m

tω ω¢ ¢

Æ [5, 6, 11], an explicit time dependence

appears in the master Eq. (3). As we will see, this approach describes correctly the coherent transport

inside the DQD but excludes the effect of the ac field on the tunneling through the contact barriers. We

did not include sources of decoherence or relaxation other than those due to the coupling to the reser-

voirs.

The current flowing through the right contact is obtained from the expresion: R

(m m mm

m m

I Γ ρ¢

, ¢

= -Â

R R1

)m m

mm m m N NΓ ρ δ

¢

¢ ¢ ¢ + ,, with an analog expresion for the current through the left contact.

R

mN is the number

of electron in the right dot when the system is in state |mÒ .

2.2 Including photon-assisted tunneling (PAT) through the contact barriers

One should expect that the applied ac field affects not only the inter-dot driving of electrons [5, 9, 10],

but also the tunneling of electrons through the leads [12, 13], as has been observed experimentally in an

ac driven single quantum dot [14]. In order to consider this effect, a re-derivation of the master equation

is needed, considering the whole time dependent Hamiltonian.

We eliminate the time dependence in the DQD energy levels applying a unitary transformation:

ac

0

ˆd ( )

ˆ ( ) e

t

t

i t H t

U t

- ¢ ¢Ú= , so that the Hamiltonian of our system becomes:

L R L R leads T

ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )H t H H H t H H t¤

= + + + + ,¢ ¢ ¢ (5)

where †ˆ ˆˆ ˆ( ) ( ) ( )O t U t OU t=¢ . Both tunneling Hamiltonians, L R

H¤

and T

ˆH , are transformed into L R

( )H t¤¢

and T

ˆ ( )H t¢ , respectively, so their matrix elements are:

L R L R

ˆ ˆ| ( ) | ( ) e | |i tm H t n J m H n

νω

ν

ν

α

•

¤ ¤

=-•

· Ò = · Ò ,¢ Â (6)

where ( )Jνα is the ν -th order Bessel function of the first kind, being

ac/Vα ω= the dimensionless ac-field

intensity, and

T T

ˆ ˆ| ( ) | e | |2

i tm H t n J m H n

νω

ν

ν

α•

=-•

Ê ˆ· Ò = · Ò .¢ Ë ¯Â (7)

Note that T

ˆ ( )H t depends on 2α / since the tunneling processes from the leads are affected by the oscilla-

tion of only one QD.

We derive the master equation for the new Hamiltonian and obtain an equation similar to that shown

in Eq. (3), substituting the inter-dot hopping, L R L R

ˆ ˆ ( )H H t¤ ¤

Æ ¢ , which is now time dependent (Eq. (6)),

and the transition rates for tunneling through the contacts, mnmn

Γ ΓÆ ¢ :

2 2

12π | | ( ( )

2 m nmn mn mn l N N

l

J fν

ν

αγ ω νω µ δΓ

•

, +

=-•

Ê ˆ= + -¢ Ë ¯Â  1

(1 ( )) )m n

nm l N Nf ω νω µ δ

, -+ - + - . (8)

3 Numerical results

3.1 Neglecting PAT through the contacts

We consider first the master equation described in Section 2.1, which does not include PAT at the con-

tact barriers, i.e., where (4) is considered for the tunneling rates through the contacts. In this case, we

phys. stat. sol. (a) 203, No. 6 (2006) 1157

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Paper

0,15 0,2 0,25 0,3ω

0

0,1

0,2

0,3

0,4

0,5

Ι/Γ

I down, non-PATI up, non-PATI down, PATI up, PAT

consider, as in previous models [5, 6, 11], that the ac field affects only the inter-dot tunneling and not the

tunneling through the contact barriers. We configure the system such that L L L

Uµ ∆> + and

R R R RU Uµ ∆< < + . The first condition allows any electron to enter the left dot to its singlet state, |

L≠ØÒ ,

while the second one allows only spin-down electrons (which have an additional Zeeman energy) to

leave the doubly occupied singlet state in the right dot towards the right lead. We consider the pumping

regime, so we choose L R

µ µ= and an asymmetric DQD such that R L

U U> .

In this configuration, if the system is initially in the state |12 |Ò = ≠Ø, ≠Ò , the current will be blocked,

unless we tune the ac-field frequency in resonance with |14 |Ò = ≠, ≠ØÒ : 2 2

14 12 LR4tω ω ,Ø

= + where

14 12 R L R LU Uω ∆ ∆

,

= - + - is the energy difference between the states |≠Ø, ≠Ò and |≠, ≠ØÒ . The ac poten-

tial delocalizes the spin down electron between the two dots via the absorption of one photon. This intro-

duces a certain probability of extracting a spin-down electron to the right lead. Then, the system acts as a

spin-down filter through the sequence | | | or | |≠Ø, ≠Ò¤ ≠, ≠ØÒÆ ≠, ≠Ò ≠Ø, ≠ØÒ Æ ≠Ø, ≠Ò [5] and the

Rabi frequency becomes renormalized as: Rabi LR

2 ( )n n

t JΩ α,

ª (where n is the number of photons in-

volved in the delocalization of the electron) [15]. Tuning the ac frequency to integer fractions of the

resonant one: nωØ/ , additional satellite peaks in the current are found due to n-photon absorption reso-

nant processes with the same spin polarization as the peak corresponding to the one photon process

(Fig. 1). Their width presents a non trivial behavior since it dependes on Rabi n

Ω,

.

The intensities of these current peaks show no significant dependence on the ac-field intensity, except

for values where the corresponding Rabi frequency vanishes, that is, for ac parameters such that

( ) 0n

J α = , where the current is quenched (see Fig. 2).

0 1 2 3 4α

0

0,1

0,2

0,3

0,4

0,5

Ι/Γ

I down, non-PATI up, non-PATI down, PATI up, PAT

Fig. 1 (online colour at: www.pss-a.com) Pump-

ed current as a function of the ac frequency.

Parameters: LR

0 005t = . , 22π| | 0 001mn

Γ γ= = . ,

L1 0U = . ,

R1 3U = . ,

L R0 026∆ ∆= = . ,

acV ω

Ø= .

Comparison of the one and two-photon resonant

current peaks (at 0 3ω ωØ

= ª . and 2ω ωØ

= / )

including PAT in the contact barriers (PAT case)

and not (non-PAT case).

Fig. 2 (online colour at: www.pss-a.com) Com-

parison of the pumped current dependence on the

ac field intensity including PAT in the contact

barriers (PAT case) and not (non-PAT case), for

fixed ac frequency 0 3ωØ= . .

1158 R. Sánchez et al.: Photon-assisted tunneling in ac driven double quantum dot spin pumps

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-a.com

3.2 Including PAT through the contact barriers

In a next step we include the effect of the ac gate voltage on the tunnelling through the contact barriers.

Even if the ac gate voltage is applied to the dots, PAT through the contacts occurs. In this case, the prob-

ability of processes taking place in the non-PAT case is reduced and new tunneling channels appear.

Now, there is a finite probability (via the absorption or emission of photons) for transitions that previ-

ously were not energetically available. Specifically, there will be processes which extract spin-up elec-

trons from the right dot doubly occupied singlet state by photon absorption through the right contact, that

is: | |R R

≠ØÒ Æ ØÒ . Since we have considered the same Zeeman splitting in both dots, both resonances

| |≠Ø, ≠Ò¤ ≠, ≠ØÒ and | |≠Ø, ØÒ¤ Ø, ≠ØÒ occur at the same frequency (which we called ωØ in the previ-

ous section). So, through the sequence | | | |≠Ø, ≠ØÒÆ ≠Ø, ØÒ¤ Ø, ≠ØÒÆ ≠Ø, ≠ØÒ, a net spin-up polar-

ized current appears (see Fig. 1). This sequence is much less efficient than the previous cycle, shown in

Section 3.1, producing spin-down current since processes that do not depend on the absorption of pho-

tons through the contacts are more probable and the system decays through the transitions

| | |≠Ø, ØÒÆ ≠Ø, ≠ØÒÆ ≠Ø, ≠Ò and | | |Ø, ≠ØÒÆ Ø, ≠ÒÆ ≠Ø, ≠Ò to the state that dominates the spin-down

cycle (shown in Section 3.1).

However, the generated spin-up current produced under PAT through the contacts, being small, can be

mostly removed as the ac-field intensity becomes small (see Fig. 2). In this case (low ac intensity), neg-

lecting PAT through the contacts is a good approximation [9], and we recover the spin-down filter be-

havior. On the other hand, the contribution of the new processes increases with the ac intensity, so the

spin-up current becomes more important, while the spin-down one is reduced in comparison with the

non-PAT case. As expected, the current vanishes as the renormalized Rabi frequency does, when

1( ) 0J α = .

This dependence of the photon-assisted spin-up current with ac

V is manifested also in the height of the

two-photons satellite peak (Fig. 1) at 2ωØ/ . The new PAT processes through the contacts increase their

rate with the ac-intensity (via the Bessel function ( 2)n

J α / , being 0n > , see Eq. (8), so they increase their

relative contribution to the current, as can be seen in the higher spin-up peak and the lower spin-down

one (Fig. 2).

4 Conclusions

We have seen that PAT through the contact barriers in ac driven DQD’s gives additional contributions to

the pumped current in comparison with the case where this effect is neglected. In particular, it modifies

the spin-down filtering behavior of a DQD device working as a spin pump by the introduction of a finite

ac-voltage dependent spin-up current. However, this contribution is effective only for high ac-intensities.

We did not include spin relaxation processes nor other effects as co-tunnelling which could be important

in the spin transport properties through the device. They will be the topic of a future work [16].

Acknowledgements Work supported by Programa de Cooperación Bilateral CSIC-CONACYT, by Grant No.

DGAPA-UNAM 114403-3, by the EU Grant No. HPRN-CT-2000-00144 and by the Ministerio de Ciencia y Tec-

nología of Spain through Grant No. MAT2002-20465.

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